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%I #19 Jan 19 2017 12:04:19
%S 1,3,57,2739,246801,35822307,7636142793,2246286827091,871869519033249,
%T 431649452286233283,265466419357802436057,198541440131880248161779,
%U 177448471205103040365902001,186781461066456338787698757027,228695537454759099917373077023593,322272887805877963568678968978370451,517868815187736150011294497645677002049
%N E.g.f. C(x)^3 = d/dx log(C(x) + S(x)), where C(x) and S(x) are described by A281181 and A281180, respectively.
%H Paul D. Hanna, <a href="/A281184/b281184.txt">Table of n, a(n) for n = 0..100</a>
%F E.g.f. C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%F E.g.f. C(x)^3 = d/dx Series_Reversion( ( sinh(x)/cosh(x)^2 + atan(sinh(x)) )/2 ).
%F E.g.f. C(x)^3 = d/dx log(C(x) + S(x)) where C(x) and S(x) satisfy:
%F (1.a) C(x)^2 - S(x)^2 = 1.
%F (1.b) C(x)^2 + S(x)^2 = 1 + Integral 4*C(x)^4*S(x) dx.
%F Integrals.
%F (2.a) S(x) = Integral C(x)^4 dx.
%F (2.b) C(x) = 1 + Integral C(x)^3*S(x) dx.
%F Exponential.
%F (3.a) C(x) + S(x) = exp( Integral C(x)^3 dx ).
%F (3.b) C(x) = cosh( Integral C(x)^3 dx ).
%F (3.c) S(x) = sinh( Integral C(x)^3 dx ).
%F Derivatives.
%F (4.a) S'(x) = C(x)^4.
%F (4.b) C'(x) = C(x)^3*S(x).
%F (4.c) (C'(x) + S'(x))/(C(x) + S(x)) = C(x)^3.
%F (4.d) (C(x)^2 + S(x)^2)' = 4*C(x)^4*S(x).
%F Explicit Solutions.
%F (5.a) S(x) = Series_Reversion( Integral 1/(1 + x^2)^2 dx ).
%F (5.b) C(x) = d/dx Series_Reversion( Integral sqrt(1 - x^2) dx ).
%F (5.c) C(x) + S(x) = exp( Series_Reversion( Integral 1/cosh(x)^3 dx ) ).
%F (5.d) C(x)^2 = d/dx Series_Reversion( Integral cos(x)^2 dx ).
%F (5.e) C(x)^3 = d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ).
%e E.g.f.: C(x)^3 = 1 + 3*x^2/2! + 57*x^4/4! + 2739*x^6/6! + 246801*x^8/8! + 35822307*x^10/10! + 7636142793*x^12/12! + 2246286827091*x^14/14! + 871869519033249*x^16/16! + 431649452286233283*x^18/18! +...
%e where related series C(x) and S(x) begin:
%e C(x) = 1 + x^2/2! + 13*x^4/4! + 493*x^6/6! + 37369*x^8/8! + 4732249*x^10/10! + 901188997*x^12/12! + 240798388357*x^14/14! + 85948640603761*x^16/16! +...+ A281181(n)*x^(2*n)/(2*n)! +...
%e S(x) = S(x) = x + 4*x^3/3! + 88*x^5/5! + 4672*x^7/7! + 454144*x^9/9! + 70084096*x^11/11! + 15728822272*x^13/13! + 4836914249728*x^15/15! + 1952137912385536*x^17/17! +...+ A281180(n)*x^(2*n-1)/(2*n-1)! +...
%e Also, the logarithm of C(x) + S(x) begins:
%e log(C(x) + S(x)) = x + 3*x^3/3! + 57*x^5/5! + 2739*x^7/7! + 246801*x^9/9! + 35822307*x^11/11! + 7636142793*x^13/13! + 2246286827091*x^15/15! +...
%e which equals Integral C(x)^3 dx.
%o (PARI) {a(n) = my(S=x, C=1); for(i=1, n, S = intformal( C^4 +x*O(x^(2*n))); C = 1 + intformal( S*C^3 ) ); (2*n)!*polcoeff(C^3, 2*n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* E.g.f. d/dx Series_Reversion( Integral 1/cosh(x)^3 dx ) */
%o {a(n) = my(C3=1); C3 = deriv( serreverse( intformal( 1/cosh(x +x*O(x^(2*n)))^3 ) ) ); (2*n)!*polcoeff(C3, 2*n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A281180 (S), A281181 (C), A281182 (C+S), A281183 (C^2).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 17 2017
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